What is the area of a polygon with vertices of (–2, 2), (3, 2), (7, –5), and (–2, –5)?
To find the area of a polygon with given vertices, we can use the Shoelace Formula.
The Shoelace Formula calculates the area of a polygon by taking the sum of the products of the coordinates of consecutive vertices, with one additional product of the last vertex with the first vertex, and subtracting the sum of the products of the coordinates of consecutive vertices in reverse order.
Given the vertices: A(-2, 2), B(3, 2), C(7, -5), and D(-2, -5), we can apply the Shoelace Formula:
Area = 1/2 * |(x1y2 + x2y3 + x3y4 + x4y1) - (x2y1 + x3y2 + x4y3 + x1y4)|
Calculating the area:
Area = 1/2 * |((-2 * 2) + (3 * 2) + (7 * -5) + (-2 * -5)) - ((2 * 3) + (2 * 7) + (-5 * -2) + (-2 * 2))|
Area = 1/2 * |(-4 + 6 - 35 + 10) - (6 + 14 + 10 - 4)|
Area = 1/2 * |(-23) - (26)|
Area = 1/2 * |-23 - 26|
Area = 1/2 * |-49|
Area = 1/2 * 49
Area = 24.5
Therefore, the area of the polygon is 24.5 square units.
Answer: 24.5 square units.
To find the area of a polygon with given vertices, you can use the Shoelace Formula (also known as the Gauss's area formula). Here are the steps to calculate it:
1. Write down the coordinates of the given vertices clockwise or counterclockwise.
In this case, the given vertices are: (–2, 2), (3, 2), (7, –5), and (–2, –5).
2. Add the products of the x-coordinates of consecutive vertices, starting from the first vertex and ending with the last vertex.
(-2 * 2) + (3 * -5) + (7 * -2) + (-2 * 2) = -4 - 15 - 14 - 4 = -37.
3. Add the products of the y-coordinates of consecutive vertices, starting from the first vertex and ending with the last vertex. (Note: Multiply the last y-coordinate with the y-coordinate of the first vertex.)
(2 * 3) + (2 * -5) + (-5 * 7) + (-2 * 2) = 6 - 10 - 35 - 4 = -43.
4. Take the absolute value of the sum of both calculations from Step 2 and Step 3.
|(-37) - (-43)| = |(-37) + 43| = |6| = 6.
5. Divide the result by 2 to get the area of the polygon.
Area = 6 / 2 = 3 square units.
Therefore, the area of the given polygon is 3 square units.
trapezoid
b1 ... (-2,2)-->(3,2)
b2 ... (-2,-5)-->(7,-5)
h ... (-2,-5)-->(-2,2)